Efficient computation of predictor derivatives

Determine whether the first- and second-order derivative quantities \(\psi_\theta(t)\) and \(\Gamma_\theta(t)\), which are required to evaluate the Laplace approximation of the marginal likelihood for the finite-dimensional MAX model, can be computed efficiently despite their reliance on filtering operations.

Background

The proposed hyperparameter-estimation procedure evaluates the Laplace approximation of the marginal likelihood. This requires the gradient-related vectors $\psi_\theta(t)$ and Hessian-related matrices $\Gamma_\theta(t)$ of the one-step-ahead predictor with respect to the truncated forward-model impulse responses. The paper derives filtering-based expressions for these quantities and shows that their construction can be organized using Hankel matrices and a number of scalar filtering operations independent of the practical truncation length $T$.

Nevertheless, the authors explicitly leave unresolved whether these derivative quantities can be computed efficiently in general, because their computation involves filtering operations. Resolving this issue would clarify the computational efficiency of the Laplace-based marginal-likelihood evaluation.